Orthogonal Nonlinear Least-Squares Regression in R

Orthogonal nonlinear least squares (ONLS) regression is a not so frequently applied and largely overlooked regression technique that comes into question when one encounters an ”error in variables” problem. While classical nonlinear least squares (NLS) aims to minimize the sum of squared vertical residuals, ONLS minimizes the sum of squared orthogonal residuals. The method is based on finding points on the fitted line that are orthogonal to the data by minimizing for each (xi, yi) the Euclidean distance ‖Di‖ to some point (x0i, y0i) on the fitted curve. There is a 25 year old FORTRAN implementation for ONLS available (ODRPACK, http://www.netlib.org/toms/869.zip), which has been included in the ’scipy’ package for Python (http://docs.scipy.org/doc/scipy-0.14.0/reference/odr.html). Here, onls has been developed for easy future algorithm tweaking in R. The results obtained from onls are exactly similar to those found in [1, 4]. The implementation is based on an inner loop using optimize for each (xi, yi) to find min ‖Di‖ within some border [xi−w, xi+w] and an outer loop for the fit parameters using nls.lm of the ’minpack’ package.